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Appl. Appl. Math.
ISSN: 1932-9466
Applications and Applied
Mathematics:
An International Journal
(AAM)
Vol. 7, Issue 1 (June 2012), pp. 71 – 98
On Lattice Structure of the Probability Functions on L*
Mashaallah Mashinchi and Ghader Khaledi
Faculty of Mathematics and Computer Sciences
Shahid Bahonar University of Kerman
Kerman, Iran
mashinchi@mail.uk.ac.ir; gh.khaledi78@gmail.com
Received: December 01, 2010; Accepted: January 02, 2012
Abstract:
In this paper, the set of all probability functions on L* is studied, where L* is the lattice of bothvalued fuzzy sets or intuitionistic fuzzy sets. It is shown that the set of all probability functions
on L* endowed with two appropriate operations has a monoid structure which is also a
distributive complete lattice. Also the lattice structure of the set of all probability functions on L*
induced by an appropriate function on [0, 1] to itself is studied. Some lattice (dual) isomorphisms
are discussed that suggests probabilities on L* could be considered in the framework of theories
modeling imprecision.
Keywords: Probability, lattice, monoid, complete lattice, fuzzy set, intuitionistic fuzzy set
MSC 2010: 06B23, 06D30
1. Introduction
Deschrijver and Kerre (2003) have shown that the underlying structure of both interval-valued
fuzzy sets and intuitionistic fuzzy sets is an L*-fuzzy set with respect to the lattice L*, in the sense
of Goguen (1967). Deschrijver and Kerre (2007) also discussed the position of intuitionistic
fuzzy set theory in the framework of theories modeling imprecision, where an overview of
interrelationships that exists between intuitionistic fuzzy set theory and other theories
modeling imprecision is described. In this direction, the study of intuitionistic
71
72
Mashaallah Mashinchi and Ghader Khaledi
balanced operators is studied by Saeb and Mashinchi (2008) which reveals an extension to
intuitionistic fuzzy set theory. A complete study of this topic is reported by Saeb (2009).
A probability p on L* has been studied by K. Lendelova and Riecan (2006). They found the
representation for a probability p on L* with respect to the Lukasiewicz connectives. Recently,
Saeb and Mashinchi (2007) followed this trend and extended the notion of a probability on a
balanced lattice, which is introduced by Homenda (2006). This topic is also considered from
different points of view by M. Rencova (2010), Riecan (2006) and Riecan and Petrovicov (2010).
The study of algebraic structures of e-implications and pseudo-e-implications on the lattice L*
are considered by Khaledi et al. (2005) and (2007). Inspired by the research on the study of
algebraic structures of implications on L*, and the direction of the study of probabilities on the
lattice L*, in this paper, the set of all probability functions on L* is considered and it is shown this
set endowed with two appropriate operations has a monoid structure which is also a distributive
complete lattice with De-Morgan algebra. Then, several other related lattice structures are
provided. The results of this paper suggest that probabilities on L* can be considered as the
representation of modeling imprecision when viewed from the perspective of Deschrijver et al.
(2007). Kaburlasos and Ritter (2007) demonstrated that lattice theory may suggest viable
alternatives in practical clustering, classification, pattern analysis and regression applications as
worthily noted by Ajmal and Jain (2009) in their recent research. The lattice structures studied in
this paper are therefore very useful apparatus in applications as explained by Ajmal, Naseem et
al. (2009) that the system of lattice algebra plays a significant role in information theory and can
be used within the numerous subfields of computational intelligence. These quotations stress that
the results reported in this paper have their potential values both from the theoretical and
application points of view in information processing.
The organization of this paper is as follows. Following this introduction some preliminaries are
discussed in Section 2. Here the structure of the lattice L* and the definition of probability on
L*are reviewed. In Section 3, the algebraic structure of the set POL, of all probabilities on L*, is
studied. In Section 4, we induce a probability function on L* by a function f : 0,1  0,1 .
Then we study the distributive complete lattice structure of the set POL f of all induced
probabilities on the lattice L*. This is done based on appropriate lattice operations on L*, when f
is a fixed strictly increasing function. Also the lattice structure of the set POL f  POL g , is
studied, where the fixed functions f and g are strictly increasing. It is proved that this structure is
a distributive complete lattice which is isomorphic to □ , where □ is the set [0,1]2 considered as
a super lattice of L*. More sub lattices of the lattices □ and L* are obtained.
2. Preliminaries
In this section, we review some known definitions and results which will be used later, for more
details see Birkhoff (1940), Deschrijver (2004) and Lendelova et al. (2006).
Definition 2.1. Let
,
| ,
0,1
and assume
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X   x1 , y 1  , Y  x2 , y2 
73
▫
.
Define
,
,
▫
,
▫
,
,
and
▫
Assume, 0
,
0,1 , 1
▫
| ,
,
.
1,0 and set
▫
0,1
1,
then, we have the following.
Theorem 2.2.
,
is a complete lattice with D as its sub lattice.
Definition 2.3. Let
,
| ,
0,1
1,
and assume
X   x1 , y 1  , Y   x 2 , y 2   L .
Define
X  L Y  Minx1 , x 2 , Max y1 , y 2 
X  L Y  Max x1 , x 2 , Miny1 , y 2 
X  L Y  x1  x 2 and y1  y 2 .
Assume, 0 L  0,1 and 1L  1,0 , then we have the following.



Lemma 2.4. L ,  L is a complete lattice.

Definition 2.5. Define the binary operations  and  on L as follows
X  Y  Min x1  x 2 ,1, Max y1  y 2  1,0
74
Mashaallah Mashinchi and Ghader Khaledi
X  Y  Maxx1  x 2  1,0, Miny1  y 2 ,1 ,
where,
X   x1 , y 1 
and
Y  x 2 , y 2  .

*
Definition 2.6. A probability on L is any function p : L  [0,1] satisfying the following
properties:
1)
p(0,1)  0 , p(1,0)  1
2)
p  X  Y   p  X  Y   p ( X )  p (Y ) for each X , Y  L
3)
If X n  X , then

for each X , X n  L , n N ,
where N is the set of natural numbers.
Remark 2.7. The notation X n  X , means that
X n  is an increasing sequence in
L and
X   Xn.
n N

Theorem 2.8. Let p : L  [0,1] be a probability on L . Then there exists
p has the following form:

 p  0,1 such that
p x, y    p x  1   p 1  y  , for all  x, y   L .
Moreover,
 p is unique.
Proof:
We only prove the uniqueness of
 p , since the rest of the proof is given by Lendelova et al.
(2006) . Suppose on the contrary that the statement is not true. So, there exist  p ,  p  0,1 ,
(1)
where
 (p1)   (p2) . Also, for all  x, y   L


p x, y    p(1) x  1   (p1) 1  y 
( 2)
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75
and


px, y    (p2) x  1   (p2) 1  y  .
Let
x0 , y0   L \ D .
So, we have:
 (p1) x0  1   (p1) 1  y 0    (p2) x0  1   (p2) 1  y 0  .
Hence,

But,
(1)
p



  (p2) x 0   (p1)   (p2) 1  y 0  .
 (p1)   (p2) , therefore, x0  1  y 0 . This is a contradiction. Hence  p is unique. ■
The following Lemma is an immediate consequence of a well-known result for sequences in ,
the fact that the limit in a cartesian product of two metric spaces ( here 0,1 ) is equal to the
pair of limits of the components, and the fact that the limit and the supremum of a sequence in a
closed subset ( here ) of a metric space is still in that subset.
*
Lemma 2.9. Let X n   xn , y n  be an increasing sequence in L . Then,


X   x, y    X n if and only if X   lim x n , lim y n  .
n N
n  
 n 
Theorem 2.10. Let
 [0,1] ,
p : L  [0,1] ,
be defined by
p  X    x1  1   1  y1  .
Then, p is a probability on
.
,
and
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Mashaallah Mashinchi and Ghader Khaledi
Proof:
Obviously
is well-defined. We show that p satisfies the conditions of Definition 2.6.
1) p(0,1)  0 , p(1,0)  1
2) Let X   x1 , y1 , Y   x 2 , y 2   L . We have:

X  Y  Minx1  x 2 ,1, Maxy1  y 2  1,0
X  Y  Maxx1  x 2  1,0, Miny1  y 2 ,1 .
Consider the following four cases:
(a) x1  x 2  1 and y1  y 2  1
In this case we have:
p  X  Y     x1  x 2   1   1  0  ,
and
p  X  Y   0   1   1   y1  y 2  .
So,
p  X  Y   p  X  Y    x1   x 2  1     1   1  y1   1    y 2
 x1  1   1  y1    x 2  1   1  y 2 
 p( X )  p(Y ) .
(b) x1  x 2  1 and y1  y 2  1 .
In this case we have:
p  X  Y    ( x1  x 2 )  1   1   y1  y 2  1 ,
and
p  X  Y    (0)  1   1  1 .
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77
So,
p  X  Y   p  X  Y     x1  x 2   1   1  y1  1  y 2 
 x1  1   1  y1    x2  1   1  y 2 
 p( X )  p(Y ) .
(c) x1  x 2  1 and y1  y 2  1
p  X  Y     1   1  0   1
p  X  Y    ( x1  x 2  1)  1   1  y1  y 2  .
So,
p  X  Y   p  X  Y   1  x1   x 2    1   1  y1   1    y 2
 x1  1   1  y1    x 2  1   1  y 2 
 p( X )  p(Y ) .
(d) x1  x 2  1 and y1  y 2  1
In this case we have: x1  x 2  y1  y 2  2 . Therefore,  x1  y1    x 2  y 2   2 . Hence,
x1  y1  1 or x 2  y 2  1 . And, X  L or Y  L . So case (d) does not occur.
Therefore, in all cases the condition 2 of Definition 2.6 does hold.
3) Let X n  X , then we have:
p X n   x n  1   1  y n  .
is an increasing sequence. Let
xn  1   1  y n   xn1  1   1  y n1  .
, so
and
.Therefore,
Hence,
.
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Mashaallah Mashinchi and Ghader Khaledi
Therefore,
lim p  X n   lim x n  1   1  y n 
n
n

  lim x n  1    1  lim y n
n 
n 
 x  1   1  y 

(By Lemma 2.9)
 p( X ) .■
3. Algebraic Structure of the Set of Probabilities on L
*
In this section we assume that X   x1 , y1  and Y   x 2 , y 2  are elements of L , unless
clearly stated otherwise.
Notation 3.1. Set



POL= p p is a probabilit y on L .
Definition 3.2. Define
 and  on POL as follows:
p  q : L  [0,1]
X  Min p ,  q x1  1  Min p ,  q  1  y1  ,
and
p  q : L  [0,1]
X  Max p , q x1  1  Max p , q  1  y1  .
Lemma 3.3. The operations  and  defined in Definition 3.2 are well-defined, closed and
associative on POL.
Proof:
It is clear that  and  are well-defined and closed on POL. We show that  and  are

associative on POL. Let p, q, r  POL and X  L .
AAM: Intern. J., Vol. 7, Issue 1 (June 2012)
  p  q   r   X   Min 
pq
79


,  r  x1  1  Min  p  q ,  r  1  y1 
 

 1  y 
, Min  ,   x  1  Min  , Min  ,    1  y 
 Min  ,   x  1  Min  ,    1  y 

 Min 
 Min Min  p ,  q  ,  r x1  1  Min Min  p ,  q  ,  r
p
p
q
qr
r
1
1
p
p
qr
q
r
1
1
1
  p   q  r   X .
Similarly, we can show that,
 p  q   r  X    p  q  r  X  .■
The following Lemma follows immediately from Theorem 2.10 by putting   1 and   0 to
obtain 1 X  and 0 X  respectively.
Lemma 3.4. Define the mappings 1,0 : L  0,1 in the following:

1 X   x1 and 0  X   1  y1 . Then, 1,0  POL .
Lemma 3.5. Let p  POL . Then:
(1)
p0  0 p  0
(2)
p 1  1 p  p
(3) p  0  0  p  p
(4) p  1  1  p  1 .
Proof:
We shall only prove (1), the other parts are similar. By commutative property of  , we have
p  0  0  p . Also,
 p  0 X   Min p ,  0 x1  1  Min p ,  0 1  y1 
 Min p ,0x1  1  Min p ,01  y1 
 1  y1
 0 X  . ■
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Mashaallah Mashinchi and Ghader Khaledi
Theorem 3.6.
,
and
,
are monoids.
Proof:
It follows from Lemmas 3.3-3.5.■
Definition 3.7. The ordering relation  on POL is defined as follows. For p, q  POL :
p  q   p q .
Definition 3.8. Define  : 0,1  POL by  a   pa , where
pa  X   ax1  1  a 1  y1  for all X  L .
The following reveals the relation between the lattices [0,1] and POL.
Lemma 3.9. Consider  in Definition 3.8, then:
(1)  is a bijection.
(2) Mina, b    a     b  .
(3) Maxa, b   a    b  .
(4) a  b if and only if  a    b  .
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81
Proof:
Straightforward. ■
Corollary 3.10 POL,  , ,   is a distributive complete lattice.
Proof:
Lemma 3.9 shows that
is an isomorphism between the lattices 0,1 and POL. Hence
POL, , ,   is a distributive complete lattice. ■
Definition 3.11. Let p  POL . We define p  in the following:
p  : L  [0,1]
p  X   1  p  X   ,
where
X    y 1 , x1  .
Lemma 3.12. Let p  POL . Then:
(i)
(ii)
(iii)
p   POL ,
 p  1   p ,

(a)  p    p ,
(b)  p , q  POL
 p  q  q 
p  ,
(iv) (De-Morgan properties):

(i )  p, q  POL   p  q   p   q  


,



(ii )  p, q  POL   p  q   p   q  


(v)
0  1 and 1  0 .
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Mashaallah Mashinchi and Ghader Khaledi
Proof:
(i)
1 p0,1  1  p1,0
 1 -1  0
p1,0   1  p 0,1
 1- 0  1
(2) We show that:
X


 Y   Y   X  and  X  Y   Y   X  .
X

 Y   Maxy1  y 2  1,0,Minx1  x 2 ,1 .
On the other hand:
Y   X   Max y1  y 2  1,0,Min x1  x 2 ,1 .
Similarly we can show that:
X

 Y   Y   X .





p  X  Y   p  X  Y   1-p  X  Y   1  p  X  Y 
 1-p Y   X    1  p Y   X  
 2-  p Y   X    p Y   X   
 2-  p Y    p  X   
 2- 1-p Y   1  p  X  
 p  X   p   Y  .
(3) Let X n  X . Then:

lim p  X n   lim 1  p X n    1  p  X   p  X  .
n 
n



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(ii)
83
p  X    p x1  1   p 1  y1 
 1 - p  X 
 1 -  p y1  1  x1   p   p x1
 1 -  p x1   p 1  y1 
So,
 p  1   p .
(iii)
(a)
 p   X   1  p  X 

 1 - 1 - p  X   



 p X 
(b) Let p  q , then  p   q . So, 1   q  1   p . Hence,
 q   p . Hence, q   p  .
(iv)
 p  q   X   1   p  q  X 
 1-Minα p ,α q y1  1  Minα p ,α q 1  x1 
 1-Minα p ,α q y1  1  x1  Minα p ,α q  Minα p ,α q x1
 Minα p ,α q 1-y1   1  Minα p ,α q x1
On the other hand:
 p  q  X   Max α p ,αq  x1  1  Max α p ,αq  1  y1  


 Max 1 - α p ,1  αq  x1  1  Max 1 - α p ,1  αq  1  y1 


 1 - Min α p ,αq  x1  Min α p ,αq  1  y1  .
Similarly, we can show that:
 p  q   p   q  .
(v)
(i)
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Mashaallah Mashinchi and Ghader Khaledi
0  X   1  0 X 
 1 - 1 - x1 
 x1  1 X 
(ii)
1 X   1  1 X 
 1-y1
 0 X  . ■
Theorem 3.13. POL,  , ,  ,0,1  is De-Morgan algebra.
Proof:
The proof follows from Lemma 3.12.■
4. f  Probability on L

In this section we induce a probability function on L by an appropriate function
f : 0,1  0,1 . Then we study the distributive complete lattice structure of the set of all

induced probabilities on L based on appropriate lattice operations, when f is a fixed strictly
increasing function.

Definition 4.1. Let f : 0,1  0,1 be any function, p : L  0,1 be a probability on L

and
 p be the unique real number obtained in Theorem 2.8. Define the induced
p f : L  0,1 by f in the following:
p f  X   f  p x1  1  f  p 1  y1  .

Lemma 4.2. The induced p f in Definition 4.1 is a probability on L .
Proof:
It is straightforward.■

In the following we will consider a class of probabilities on L induced by Sugeno negation.
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85

Example 4.3. Let p : L  0,1 be a probability function on L . Consider the Sugeno

negation N  : 0,1  0,1 , where
N  x  
1 x
,   1,   .
1  x
Then,
 1 p
p N  X   
 1  
p


 1    p
 x1  

 1  
p



is a probability on L , where

1  y1 ,   1,  


 p is the unique real number obtained in Theorem 2.8.

Lemma 4.4. Let p : L  0,1 be an arbitrary probability function on L and f : 0,1  0,1

be any function. Then p f in Definition 4.1 is onto.
Proof:

Let y  0,1. Define X   y ,1  y  . It is clear that X  L and p f  X   y .■
p : L  0,1 be an arbitrary probability function on L and
f : 0,1  0,1 be any function. Then p f in Definition 4.1 is not 1-1.
Remark 4.5. Let

Define X  1 f
and
 , f  , where 
p
p
0,0 . It is clear that ,
p
is the unique real number obtained in Theorem 2.8
and X  Y . Also, p f  X   p f Y   1  f
  .
p
Notation 4.6. Let f : 0,1  0,1 be any function. Set:
POL f  p f p  POL.
Definition 4.7. Let POL f be as in Notation 4.6. Define the operations  and  on POL f
as follows:
p f  q f : L  0,1
X  Minf α p ,f α q x1  1  Minf α p ,f α q 1  y1  ,
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Mashaallah Mashinchi and Ghader Khaledi
and
p f  q f : L  0,1
X  Maxf α p ,f α q x1  1  Maxf α p ,f α q 1  y1  .
Definition 4.8. For a fixed f : 0,1  0,1 , define the ordering relation  f on POL f as
follows:
p f  f q f  f  p   f  q  , for all p f , q f  POL f .
Lemma 4.9. Let f : 0,1  0,1 be a strictly increasing (strictly decreasing) function and
p, q  POL . Then:
(1)


(2)


.
Proof:

(1) Let X  L and f : 0,1  0,1 be a strictly increasing function, then:
( p  q) f  X   f  p  q x1  1  f  p  q 1  y1 




 f Min p ,  q  x1  1  f Min p ,  q  1  y1 
 Minf  p , f  q x1  1  Minf  p , f  q 1  y1 
 ( p f  q f ) X  .

Let X  L and f : 0,1  0,1 be a strictly decreasing function then:
( p  q) f  X   f  pq x1  1  f  pq 1  y1 




 f Min p ,  q  x1  1  f Min p ,  q  1  y1 
 Maxf  p , f  q x1  1  Maxf  p , f  q 1  y1 
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87
 ( p f  q f ) X  .
(2) The proof is similar to part (1). ■
Definition 4.10. Let f : 0,1  0,1 be a function. Define  : POL  POL f as
  p  p f .
Lemma 4.11. Let f : 0,1  0,1 be a function, p, q  POL and  be as in Definition 4.10,
then:
(1)  is well-defined.
If f : 0,1  0,1 is strictly increasing (strictly decreasing) function then:
(2)  is a bijection,
(3)   p  q     p  
 q  (  p  q     p    q  ),
(4)   p  q     p    q  (  p  q     p  
 q  ),
(5) p  q if and only if   p   f
 q 
( p  q if and only if  q   f   p  ).
Proof:
It is similar to the proof of Lemma 4.9. ■
Now the following fact is immediate.
Corollary 4.12. Let f : 0,1  0,1 be a function.

(1) If f is a strictly increasing function, then POL f ,  ,  ,  f
 is a distributive complete
lattice.
(2) If f is a strictly decreasing function, then  in Definition 4.10 is a dual isomorphism.
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Mashaallah Mashinchi and Ghader Khaledi
Example 4.13.
(1) Consider
: 0,1
0,1 , where
, where
1

1
Hence, POL f ,  ,  ,  f
(2) Consider
and
: 0,1
, where
1
Hence,
. Then
is a strictly increasing function and
.
 is a distributive complete lattice.
0,1 , where
1
1
. Then
is a strictly decreasing function
.
is a dual isomorphism.
Remark 4.14. As Birkhoff, G. (1940) mentioned, the product of two lattices is also a lattice. Let
f , g : 0,1  0,1 be two strictly increasing functions, then POL f  POL g is also a
distributive complete lattice.
Lemma 4.15. Let f : 0,1  0,1 be onto. Then POL  POL f .
Proof:
It is clear that POL f  POL . Let p  POL . We show that p  POL f . Since p  POL ,
we have:
p X    p x1  1   p 1  y1  and  p  0,1 .
f is onto, so there exist
  0,1 such that f     p . Define
q  X    x1  1   1  y1  .
By Theorem 2.10, q  POL and q f  X   f  x1  1  f  1  y1  . Therefore
q f  X    p x1  1   p 1  y1   p X  .
Hence, p  POL f . Therefore, POL  POL f .■
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Corollary 4.16. Let N  be the Sugeno negation as in Example 4.3, then POL  POL N  .
Definition 4.17. Let f , g : 0,1  0,1 be two functions. Define
:
▫
by
  ,     p f , q g  ,
where, for all
p f  X   f  x1  1  f  1  y1  ,
and
q g  X   g  x1  1  g  1  y1  .
Definition 4.18. Let f , g : 0,1  0,1 be two functions and assume
p
f
, q g  , r f , s g  POL f  POL g . Define
(1)
,
▫
,
 rf , qg  s g  ,
(2)
,
▫
,
 rf , qg  s g ,
(3)
,
,
▫
if and only if p f  f r f and q g
g
 sg ,
Theorem 4.19. Let f , g : 0,1  0,1 be two functions and  be as in Definition 4.17. Then:
(1)  is well-defined.
If f , g : 0,1  0,1 be two strictly increasing (strictly decreasing) functions then:
(2)  is a bijection.
(3)
(
(4)
(
,
▫
,
,
▫
,
,
▫
,
,
▫
,
,
▫
,
,
▫
,
,
▫
,
,
▫
,
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Mashaallah Mashinchi and Ghader Khaledi
,
(5)
,
▫
,
▫
if and only if
,
if and only if
,
,
▫
,
▫
.
,
.)
Proof:
,
Let
,

and X  L . Define p f , q g , r f and s g as follows:
,
p f  X   f  x1  1  f  1  y1  ,
q g  X   g  x1  1  g  1  y1  ,
r f  X   f  x1  1  f  1  y1  ,
and
s g  X   g  x1  1  g  1  y1  .




It is clear that p f , q g and r f , s g  POL f  POL g .
(1) Let
So p f
 ,     ,   , therefore    and    . Hence f    f   and g    g   .
 r f and q g  s g . Therefore,   ,      ,  .
We only prove the Lemma in the case that f, g are strictly increasing functions. The proof of the
case that f, g are strictly decreasing functions is similar.
  ,      ,  . Hence  p f , q g   r f , s g . Therefore, p f  r f and q g  s g . So
f    f   and g    g   . Hence,    and
. Therefore,  ,     ,  .
(2) Let


Let u f , v g  POL f  POL g , such that:
u f  X   f  u x1  1  f  u 1  y1 
and
v g  X   g  v x1  1  g  v 1  y1  .
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,
It is clear that
(3)
91
  u , v   u f , v g  .
and
  ,    L  ,    Min ,  , Max ,   m f , n g  ,
where,
,
1
,
1
and
n g  X   g Max ,  x1  1  g Max ,  1  y1  .
Therefore,
1
,
,
1
,
and


ng  X   Max  g    , g   x1  1  Max  g    , g   1  y1  .

So, m f  X   p f  r f
 X  and n  X   q
g
g
 s g  X  .
Hence,
  ,    L  ,    p f  r f , q g  s g 
,
▫
,
▫
,
,
.
(4) It is similar to the part (3).
(5) 
Let
,
▫
,
. Hence,
   and    . So, f    f   and g    g   .
Therefore, p f  f r f and q g

It is similar to the above part. ■
g
 s g . Hence,
,
▫
,
.
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Mashaallah Mashinchi and Ghader Khaledi
Corollary 4.20. Let f , g : 0,1  0,1 be two functions.
(1) If f and g are two strictly increasing functions, then POL f  POL g is a distributive
.
complete lattice which is isomorphic to
(2) If f and g are two strictly decreasing functions, then  in Definition 4.17 is a dual
isomorphism.
Example 4.21.
(1) Consider , : 0,1
functions and
0,1 , where
,
1
1
1
1
,
,
. Then ,
are strictly increasing
, where
and
.
Hence, POL f  POL g is a distributive complete lattice which is isomorphic to L .
(2) Consider , : 0,1
decreasing functions and
0,1 , where
1
,
, where
,
1
1
1
1
,
and
Then
.
is a dual isomorphism.
Notation 4.22. Let f , g : 0,1  0,1 be two functions. Define
L POL f POL g  POL f  POL g as follows:
L POL f POL g   p f , q g  p, q  POL and  p   q  1.
1
. Then ,
are strictly
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Definition 4.23. Let f , g : 0,1  0,1 be two functions. Define  : L  L POL f POL g by

 ,     p f , q g , where
p f  X   f  x1  1  f  1  y1 
and
q g  X   g  x1  1  g  1  y1  , for all X  L .
Definition 4.24. Let f , g : 0,1  0,1 be two functions and assume
p
, q g  , r f , s g  L POL f POL g .
f
Define:



 




 





(1) p f , q g  L r f , s g  p f  r f , q g  s g
(2) p f , q g  L r f , s g  p f  r f , q g  s g
(3) p f , q g  L r f , s g if and only if p f  f r f and q g
g
 sg .
Theorem 4.25. Let f , g : 0,1  0,1 be two functions and  be as in Definition 4.23. Then:
(1)  is well-defined.
If f , g : 0,1  0,1 are strictly increasing (strictly decreasing) functions then:
(2)  is a bijection.

(3)   ,    L
 ,    ,    L  , 

 ,     ,    ,     , 
L

(4)   ,    L
L
 ,    ,    L  , 

 ,     ,    ,     , 
L
L
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Mashaallah Mashinchi and Ghader Khaledi
(5)  ,    L
 ,   if and only if  ,    L  ,  .

 ,     ,  if and only if  ,    ,   .
L
L
Proof:
It is similar to the proof of Theorem 4.19. ■
Corollary 4.26 Let f , g : 0,1  0,1 be two functions.
(1) If f and g are two strictly increasing functions, then L POL f POL g is a distributive complete

lattice which is isomorphic to L .
(2) If f and g are two strictly decreasing functions, then  in Definition 4.23 is a dual
isomorphism.
Example 4.27.
(1) Consider ,
in Example 4.21 part (1). Then
L POL f POL g   p f , q g  p, q  POL and  p   q  1,
where
1
1
1
1
and
.

Hence, L POL f POL g is a distributive complete lattice which is isomorphic to L .
(2) Consider ,

1
1
1
1
and
Then,

in Example 4.21 part (2). Then  ,    p f , q g , where
is a dual isomorphism.
.
AAM: Intern. J., Vol. 7, Issue 1 (June 2012)
95

Definition 4.28. Let f , g : 0,1  0,1 be two functions. Define D POL f POL g  p f , q g
p, q  POL and  p   q  1.

Lemma 4.29. Let f , g : 0,1  0,1 be two functions. Then
D POL f POLg   p f , pg  p  POL.
Proof:


Let p f , q g  D POL f POL g . So,
 p   q  1 . Hence,  q  1   p . By Lemma 3.12, we have
q  p  . Therefore, q g  p g . Hence,
p  POL.
Let
p
, pg   p f , pg 
f
p
p  POL .
f
, q g    p f , p g  . So,
By
Lemma
 p   p   p  1   p  1 . Therefore,  p f , p g   D POL
f
3.12,
POL g
p
f
, q g   p f , pg 
 p  1   p ,
so
.■
Definition 4.30. Let f , g : 0,1  0,1 be two functions.


Define  : D  D POL f POL g , by   ,1     p f , p g , where
p X   x1  1   1  y1  for all X  L .
Theorem 4.31. Let f , g : 0,1  0,1 be two functions and  be as in Definition 4.30. Then:
(1)  is well-defined.
If f , g : 0,1  0,1 are two strictly increasing (strictly decreasing) functions, then:
(2)  is a bijection.
(3) 
 ,1      ,1      ,1    
L
L
 ,1      ,1      ,1    
L
(4) 
 ,1      ,1      ,1    
L
  ,1   
L
  ,1   
L
  ,1   
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Mashaallah Mashinchi and Ghader Khaledi
 ,1      ,1      ,1    
L
(5)  ,1     L
  ,1   
L
 ,1    if and only if  ,1     L

 ,1      ,1    if and only if  ,1    
L
 ,1    .
L
 ,1    .
Proof:
The proof is similar to the proof of Theorem 4.19. ■
Corollary 4.32 Let f , g : 0,1  0,1 be two functions.
(1) If f and g are two strictly increasing functions, then D POL f POL g is a distributive complete
lattice which is isomorphic to D .
(2) If f and g are two strictly decreasing functions, then  in Definition 4.30 is a dual
isomorphism.
Example 4.33.
(1) Consider ,
in Example 4.21 part (1). Then
D POL f POLg   p f , pg  p  POL,
where
1
1
and
́
1
1
1
.
1
Hence, D POL f POL g is a distributive complete lattice which is isomorphic to D .
(2) Consider ,

1
1
and
́
1

in Example 4.21 part (2).Then   ,1     p f , p g , where
1
1
1
.
AAM: Intern. J., Vol. 7, Issue 1 (June 2012)
97
 is a dual isomorphism.
Remark 4.34. Note that the results given in Corollary 4.26 show that a probability on the lattice
L POL f POL g (as an isomorphism of L ) could be viewed as a representation of modeling
imprecision, if it is seen from the perspective of Figure 1 in the paper of Ajmal, Naseem et al.
(2009) , where it is proved that different models of imprecision such as grey sets, vague sets,

intuitionistic [0,1]-fuzzy sets, L - fuzzy sets, intuitionistic fuzzy sets and interval-valued fuzzy
sets are equivalent up to isomorphism.
5. Conclusion
In this paper, POL, the set of all probability functions on L* is studied. It is shown that this set is
sufficiently large by constructing many examples using Sugeno’s negation. Two operations are
defined to endow this set as monoid structure which is a distributive complete lattice and also
De-Morgan algebra. Then, POL f , the set of all f  probabilities on L*, induced by a fixed
strictly increasing function on [0,1] to itself is studied and it is proved that this set is a
distributive complete lattice when endowed with appropriate lattice operations. It is shown that
the product lattice POL f  POL g , when f and g are strictly increasing functions, is a
distributive complete lattice isomorphic to , where is the set [0,1]2 considered as a super
and L*are obtained. Some lattices (dual) isomorphism
lattice of L*. Then more sub lattices of
studied in this paper actually reveal that probabilities on the lattice L*could be considered as a
representation of modeling imprecision as explained in Remark 4.34.
Acknowledgments
This research is supported by a grant from Mahani Mathematical Research Center at Shahid
Bahonar University of Kerman, Iran. The authors also would like to thank Professors M.
Rencova and B. Riecan for sending their papers.
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